converging theoretical perspectives on charge pumping

Slava Kashcheyevs

Colloquium at Physikalisch-Technische Bundesanstalt (Braunschweig, Germany)

November 13th, 2007

Converging theoretical perspectives on charge pumping

Pumping: definitions

• rectification• photovoltaic effect• photon-assisted tunneling• ratchets

Pumping overlaps with:

Interested in “small”pumps to witness:• quantum interference• single-electron

charging

f

I

Outline• Adiabatic Quantum Pump

Thouless pumpBrouwer formulaResonances and quantization

• Beyond the simple pictureNon-adiabaticity (driving fast)Rate equations and Coulomb interaction

• Single-parameter, non-adiabatic, quantized

Adiabatic pump by Thouless

• If the gap remains open at all times,I = e f (exact integer)

• Argument is exact for an infinite system

DJ Thouless, PRB 27, 6083 (1983)

Adiabatic Quantum Pumping

a phase-coherent conductor

Pump by deforming

Change of interference pattern can induce “waves” traveling to infinity

Brouwer formula gives

I in terms of

Brouwer formula: “plug-and-play”

Vary shape via X1(t), X2(t), ..

Solve for “frozen time”scattering matrix

Brouwer formula gives

I in terms of

Brouwer formula: “plug-and-play”

Brouwer formula gives

I in terms of

Brouwer formula: “plug-and-play”

Brouwer formula gives

I =

• Depends on a phase• Allows for a

geometric interpretation

• Need 2 parameters!

“B”

Outline• Adiabatic Quantum Pump

Thouless pumpBrouwer formulaResonances and quantization

• Beyond the simple pictureNon-adiabaticity (driving fast)Rate equations and Coulomb interaction

• Single-parameter, non-adiabatic, quantized

• Idealized double-barrier resonator

• Tuning X1 and X2 to match a resonance• I e f, if the whole resonance line

encircled

Resonances and quantization

Y Levinson, O Entin-Wohlman, P Wölfle Physica A 302, 335 (2001)

X1 X2

• How can interference lead to quantization?

• Resonances correspond to quasi-bound states

• Proper loading/unloading gives quantization

X1 X2

Resonances and quantization

V Kashcheyevs, A Aharony, O Entin-Wohlman, PRB 69, 195301 (2004)

Outline• Adiabatic Quantum Pump

Thouless pumpBrouwer formulaResonances and quantization

• Beyond the simple pictureNon-adiabaticity (driving fast)Rate equations and Coulomb interaction

• Single-parameter, non-adiabatic, quantized

Driving too fast: non-adiabaticity• What is the meaning of “adiabatic”?

• Can develop a series:

• Q: What is the small parameter?

Thouless:staying in the ground state

Brouwer:a gapless system!O Entin, A Aharony, Y Levinson

PRB 65, 195411 (2002)

M Moskalets, M Büttiker PRB 66, 205320 (2002)

Floquet scattering for pumps• Adiabatic scattering matrix S(E; t)

is “quasi-classical”

• Exact description by

• Typical matrix dimension(# space pts) (# side-bands) LARGE!

h f

M Moskalets, M Büttiker PRB 66, 205320 (2002)

Adiabaticity criteria• Adiabatic scattering matrix S(E; t)

• Floquet matrix

• Adiabatic approximation is OK as long as ≈ Fourier T.[ S(E; t)]

• For a quantized adiabatic pump, the breakdown scale is f ~ Γ (level width)

h f

Outline• Adiabatic Quantum Pump

Thouless pumpBrouwer formulaResonances and quantization

• Beyond the simple pictureNon-adiabaticity (driving fast)Rate equations and Coulomb interaction

• Single-parameter, non-adiabatic, quantized

• A different starting point

• Consider states of an isolated, finite device

• Tunneling to/from leads as a perturbation!

Rate equations: concept

ΓRΓL

Rate equations: an example

For open systems & Thouless pump, see GM Graf, G Ortelli arXiv:0709.3033

ε0- i (ΓL

+ΓR)

• Loading/unloading of a quasi-bound state

• Rate equation for the occupation probability

• Interference in an almost closed systemjust creates the discrete states!

• Backbone of Single Electron Transistor theory

• Conditions to work: Tunneling is weak: Γ << Δε or Ec No coherence between

multiple tunneling events: Γ << kBT

• Systematic inclusion of charging effects!

Rate equations are useful!

DV Averin, KK Likharev “Single Electronics” (1991)

CWJ Beenakker PRB 44, 1646 (1991)

Outline• Adiabatic Quantum Pump

Thouless pumpBrouwer formulaResonances and quantization

• Beyond the simple pictureNon-adiabaticity (driving fast)Rate equations and Coulomb interaction

• Single-parameter, non-adiabatic, quantized

Single-parameter non-adiabatic

quantized pumping

B. Kaestner, VK, S. Amakawa, L. Li, M. D. Blumenthal, T. J. B. M. Janssen, G. Hein, K. Pierz, T. Weimann, U. Siegner,

and H. W. Schumacher, arXiv:0707.0993

“Roll-over-the-hill”

V2(mV)

• Fix V1 and V2

• Apply Vac on top of V1

• Measure the current I(V2)

V1

V2

V1 V2

Experimental results

A simple theory

ε0

• Given V1(t) and V2 , solve the scattering problem

• Identify the resonanceε0(t) , ΓL (t) and ΓR (t)

• Rate equation for the occupation probability P(t)

A: Too slow (almost adiabatic)

Enough time to equilibrate

Charge re-fluxes back to where it came from → I ≈ 0

ω<<Γ

B: Balanced for quantization

Tunneling is blocked, while the left-right symmetry switches to opposite

Loading from the left, unloading to the right

→ I ≈ e f

ω>>Γ

C: Too fast

Tunneling is too slow to catch up with energy level switching

The chrage is “stuck” → I ≈ 0

ω

A general outlook

I / (ef)